The Inclination of the Planetary System Relative to the Solar Equator May Be

Home , Invariable plane

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Planetary system ment. More recently, Batygin and Brown (2016) formation predicts that planets are formed from a studied more deeply the orbital alignment of those disk of gas and dust and this disk rotates on the distant TNOs, showing that the six most distant same plane of the star’s equator. The final plan- objects exhibit also a clustering in their longi- etary orbits, if no mutual close encounters take tudes of node; they estimated that the probability place, must be approximately coplanar and coin- that this double alignment in argument of perihe- cident with the star’s equator. We thus suppose lion and longitude of the node is just fortuitous that the giant planets and the solar equator were is 0.007%. Moreover they showed that a planet 9 initially on the same plane. We assume that pl9 (hereafter named just pl9) could account for said was scattered away from the region of the other alignment if it had a mass of about 10M⊕ and an giant planets when the disk was still present and orbit with semimajor axis between 300 and 900 the solar system was still embedded in a stellar au, perihelion distance between 200 and 350 au, cluster (Izidoro et al. 2015). The stellar cluster is and orbital inclination of about 30◦ to the ecliptic needed, so that the perihelion of the orbit of pl9 plane. The Batygin-Brown approach based on sec- can be lifted and pl9 can decouple from the other ular dynamics is able to determine an approximate planets (Brasser et al. 2008). Because most of the orbit for the distant planet that could explain the angular momentum is in the protoplanetary disk, said alignment, but not the planet’s position on it is likely that the ejection of pl9 onto an inclined that orbit. Fienga et al. (2016) use a typical or- orbit did not significantly change the inclination bit among those proposed by Batygin and Brown of the disk and of the other giant planets. Notice (2016) and determined the range in true longi- also that the inclination of pl9 might have been tude of pl9 on that orbit that decreases the resid- increased by the action of the cluster, while lift- uals in INPOP ephemerids of Saturn, relative ing the perihelion in a Lidov-Kozai like dynamics to the Cassini data. Holman and Payne (2016) (Brasser et al. 2008). Thus, we assume that, at obtained a similar result using JPL ephemerids. the removal of the protoplanetary disk and of the Brown and Batygin (2016) refined their previous birth cluster of the Sun, the 4 major giant planets results by further constraining the mass and or- were on orbits near the solar equator, while pl9’s bital elements of pl9 that are compatible with the orbit was off-plane. At this point, a slow preces- observed TNOs orbital alignment. They now ar- sion of iv4 started to take place relative to the gue for pl9’s semimajor axis in the range 380980au, total angular momentum vector of the Solar Sys- perihelion distance in the range 150350 au and a tem, including pl9, keeping however the orienta- mass between 5 and 20M⊕, for an orbital inclination of the solar equator plane unchanged. Thus tion of 30◦. Malhotra et al. (2016) looked for extra the current angle between the solar equator and constraints on pl9 orbit by analyzing the orbital iv4 (about 6◦ - see below) must be a signature of periods of the four longest period TNOs. Their pl9 perturbation and we aim at finding ranges of approach is based on the supposition that pl9 is in orbital elements and mass for pl9 that can explain mean motion resonances with those TNOs. Beust quantitatively the present tilt of iv4 relative to the (2016), however, showed that a mean motion reso- solar equator. nant configuration is not necessary to explain the The solar equator with respect to the eclip- ◦ orbital confinement. tic is identified by an inclination IS = 7.2 and ◦ Here we study the precession of the plane or- a longitude of the ascending node ΩS = 75.8 thogonal to the total angular momentum of the (Beck and Giles 2005). The invariant plane with four giant planets due to the perturbation of pl9. respect to the ecliptic is defined by an inclination ◦ We find that, given the large distance of pl9, the Ii = 1.58 and a longitude of the ascending node ◦ dynamics of the giant planets can be decomposed Ωi = 107.58 (Souami and Souchay 2012). Em- into a classic Lagrange-Laplace dynamics relative ploying two rotations we can find the invariant to their own invariant plane (the plane orthogonal plane angles with respect to the solar equator to ◦ ◦ to their total angular momentum vector, hereafter be Iv = 5.9 and Ωv = 171.9 . We will use these

2 parameters throughout the rest of the paper. We also consider that iv4, as above defined, is equiv- N alent to the invariant plane of the solar system −nj mk (1) Bjj = αjkα¯jkb3/2(αjk) 4 M⊙ + mj mentioned in the works above, which take into ac- k=1X,k=6 j count also the inner planets. nj mk (1) Bjk = αjkα¯jkb3/2(αjk ) (2) In Section 2 we develop two analytical ap- 4 M⊙ + mj proaches aimed at determining the tilt experienced by iv4 due to the perturbation of pl9. We also where M⊙ is the mass of the Sun, mj and mk perform some numerical integrations of the full are the masses of the interacting bodies, and nj equations of motion to validate our analytical ap- is the mean motion of the planet j. αjk = aj /ak proaches. In Section 3, we apply our analytical andα ¯jk = αjk for aj < ak. For aj > ak we have (1) method to determine the range of masses and or- αjk = ak/aj andα ¯jk = 1. b3/2(αjk) is the Laplace bital elements of pl9 that can account for the ob- coefficient of the first kind (Murray and Dermott served tilt of iv4 to the solar equator plane. In 1999, Ch. 7). Section 4, we draw our conclusions. In the context of the Laplace-Lagrange secular theory, valid for small values of the inclina- 2. Methods tion and eccentricity, if we want to account for large values of the inclination and eccentricity of We first apply the classical Laplace-Lagrange pl9 we need to add some new ingredients to the formalism up to second order in the inclination classical theory. In this manner, the inclination to evaluate the variation of the inclination experi- of pl9 was accounted for by reducing pl9’s mass enced by iv4 to due pl9. Since a second order ap- by a factor of sin I, i.e. m = m cos I . proach may not be sufficient for large inclinations 9(new) 9(real) 9 By doing this, we consider only the projection of of pl9, we develop another approach based on the the mass of pl9 onto the planet’s reference frame angular momenta of pl9 and the four giant plan- (iv4) (Batygin et al. 2011). As for the eccentricity ets. In this case, we make no approximation on of pl9, assuming that one cannot derive a simple the inclinations but just a first order approxima- secular approach (Murray and Dermott 1999), it tion in the ratio of the known planets’ semimajor turns necessary to somehow incorporate the aver- axes to that of pl9. aged effect of an eccentric orbit upon the motion of 2.1. First approach: secular perturbations the perturbed planet. According to Gomes et al. to second order (2006), the averaged effect can be computed assuming that the perturber is on a circular orbit of − 2 Following Batygin et al. (2011), we derive a sec- radius b, where b9 = a9 1 e9 is the semi minor ular theory of the evolution of the inclination of axis of the real perturber’sp orbit. Thus, in order iv4 based on the classical Laplace-Lagrange theory to compute for the possible large eccentricity of − 2 up to the second order in the inclinations. From pl9 we will assume b9 = a9 1 e9 as its circular Murray and Dermott (1999), we have the follow- semimajor axis analog. p ing form for the classical Hamiltonian: Therefore, with the implementations of m9(new) = − 2 m9(real) cos I9 and b9 = a9 1 e9, it is possible N N to rewrite the Hamiltonianp (1) in terms of the 1 H = B I I cos(Ω − Ω ) (1) vertical and horizontal components of the inclina- 2 jk j k j k Xj=1 kX=1 tion (pj = Ij sin Ωj and qj = Ij cosΩj ), where the first-order perturbation equations (p ˙j = ∂Hj /∂qj where j and k indicate the perturbed and the per- andq ˙j = −∂Hj/∂pj) lead to an eigensystem that turbing bodies respectively. The Nth index refers can be solved analytically (Murray and Dermott to pl9. All inclinations are expressed with respect 1999, Ch. 7) to the solar equator, supposed as the initial fixed reference frame. The coefficients Bjj and Bjk as- N sume the form: pj = Ijk sin(fkt + γk) kX=1

3 N fast variables for both planets. For that we sup- qj = Ijk cos(fkt + γk) (3) pose two reference frames defined on each of the kX=1 planets’ orbits making an angle I between them. ~ ~ ~ ~ ~ ~ with f being the set of N eigenvalues of matrix B The frames are defined by (i, j, k) and (i9, j, k9), k ~ (Eq. 2), I the associated eigenvector, and γ a unitary vectors where the component j is common jk k ~ phase angle determined by the initial conditions. to both frames. j is in the intersection of the or- This leads to the final solution bital planes and lies on the invariant plane defined by both planets; ~i and ~i9 are orthogonal to ~j on each of the orbital planes and ~k and ~k9 completes 2 2 the reference frames through the right hand rule. Ij = pj + qj q It must be noted that these frames are defined pj ~ Ωj = arctan( ). (4) just to compute the derivative of h on those com- qj ponents instantaneously. We now assume that the perturbed planet has a small enough eccentricity Finally, starting with Jupiter, Saturn, Uranus, so as to consider its orbit as circular. On the other and Neptune in the equatorial plane of the Sun, hand, the perturbing planet will be considered ec- given the orbital parameters of pl9, one can ver- centric. In this manner we can represent the radius ify that the eigenvectors (Ij5, j = 1...4) have the vector of each of the planets as: same magnitude. In this way, the equations in (4) also represent the evolution of the pair (I,Ω) ~r = (a cos l)~i + (a sin l) ~j (7) of iv4. Despite the modifications we introduced to account for the large inclination and eccentricity of pl9, our method has limitations, being less ~r9 = (r9 cos θ9)~i9 + (r9 sin θ9) ~j , (8) accurate for large values of I9 and e9. where a is the perturbed planet semimajor axis, 2.2. Second approach: angular momen- l is the perturbed planet mean longitude and θ9 tum is the angle from the intersection of the planes to the perturbing planet’s position, which is the sum The equation of motion of a planet around a of pl9’s true anomaly f and the longitude of the star perturbed by a second planet in a reference 9 ascending node with respect to the invariant plane. frame centered in the star is: We now put together Eq. 6, 7 and 8 and develop the vectorial products remembering that ~i ×~i9 = − sin I~j, ~i×~j = ~k, ~j ×~i = −~k = − sin I~k +cos I~i ¨ − ~r ~r19 − ~r9 9 9 ~r = G(m9 + M) 3 + Gm9 3 3 (5) and ~j×~j = 0. Developing the components in~i and r r19 r9 ~k, we notice that there is always a trigonometric where the subscript 9 refers to the perturbing function in at least one of the fast angles to an odd planet and the perturbed planet has no subscript. power, which results in a null average for these In this equation, ~r is the radius vector and r its components. absolute value, m9 is the perturbing planet mass, For the ~j component, after developing the vec- M the star’s mass, G the gravitational constant torial product to the first order in a/r9, we obtain: and r19 the distance between both planets. Let us define the angular momentum per unit mass by ~ × ˙ ~ × 3 − −3 h = ~r ~r. Using Eq. 5, the time derivative of h (~r ~r9)/r19 = (a r9 cos l cos θ9 sin I) r9 T (9) can be found as: where, ˙ ~r × ~r19 ~r × ~r9 ~h = Gm − Gm (6) 9 3 9 3 3 a2 a r19 r9 − T =1 2 +3 (sin l sin θ9 + cos l cos θ9 cos I) 2 r9 r9 Since ~r19 = ~r9 −~r and ~r×~r = 0, we have to deal × just with the vectorial product ~r ~r9 in Eq. 6. We The first order approximation can be quite ac- now want to average the right hand of Eq. 6 in the curate when a/r9 is small, which is the case of a

4 distant planet perturbing a close in one. Averag- 0 ing in the fast angles l and f9, which appears in I9=30 θ9 and r9, for one orbital period, we arrive at: angular momentum numerical integration 7 secular theory P P9 × 1 ~r ~r9 2 3 2 −3 3 dt = a b9 sin 2I (10) P P9 Z0 Z0 r19 8 5 where P and P9 are the orbital periods of the perturbed and perturbing planets, respectively, and I (deg.) ∆ − 2 3 b9 = a9 (1 e9) is the semiminor axis of the per- × 3 turbingp planet. The term (~r ~r9)/r9 averages to zero in all components. The variation of ~h be- comes: 1

˙ 3 2 −3 400 600 800 1000 ~h = Gm9 a b sin 2I~j (11) 8 9 semimajor axis (au) Through the way the reference frames were con- Fig. 1.— The inclination of iv4 relative to the structed, we have shown that ~h has only and al- initial plane (the solar equator) after 4.5 Gy due ways a non-zero time derivative in the direction to the presence of pl9 with e =0.7, as a function of the intersections of the orbital planes. Since 9 of pl9’s semimajor axis. The dashed curve shows the choice of the reference frames could be for any the prediction from the secular theory described time, we conclude that at any time the non-zero in sect. 2.1 and the solid curve from the analytic component of the time derivative of ~h is orthogonal theory described in sect. 2.2. The dots show the to ~h on the intersection of the orbital planes. This results of 6 numerical simulations that contain no is satisﬁed only if the projection of ~h on the invari- approximations. ant plane is a circle around the origin. The radius of the circle α (where sin α = H9/Ht sin I) is the angle between ~h and H~ t where H~ t = m~h + H~ 9 is ~ × ˙ 0 the total angular momentum and H9 = m9~r9 ~r9 I9=30 is pl9 angular momentum. The precession fre- angular momentum quency is the coeﬃcient multiplying ~j in Eq. 11 numerical integration divided by 2παh, where h is the absolute value of secular theory ~h = GM⊙a. 270 Thisp approach to compute the variation of the inclination and node of a planet perturbed by an- 180 other one can be extrapolated to the case of a (deg.)

distant planet perturbing several close in planets. ∆ Ω We noticed by numerical integrations that this approach is accurate enough for the Solar System gi- 90 ant planets perturbed by a distant planet, when we replace the four planets by only one with semimajor axis at 10.227 au and the same angular momentum as the resultant angular momentum of 400 600 800 1000 semimajor axis (au) the giant planets. Fig. 2.— the same as Fig. 1 but for the longitude 2.3. Comparison of both approaches with of the node of iv4 relative to the solar equator. numerical integrations Here the two analytic approaches look indistin- We considered the current orbital elements of guishable the four giant planets and a pl9 and ran a numer-

5 ical integration of the full equations of motion for 4.5Gy. We noticed that iv4 behaved the same way as if all planets started on the same plane. Thus e9=0.7

for simplification we started new integrations with I =15o 9 o the current giant planets orbital elements except I9=30 I =45o 10 9 o for the inclinations which were started at zero and I9=60 these integrations were used for the comparisons below. We ran a total of six different numerical integrations with different semimajor axes for pl9 1. 7

The planet’s mass and other orbital elements are I (deg.)

−5 ◦ ◦ ∆ m9 =3 × 10 M⊙, e9 =0.7, I9 = 30 ,Ω9 = 113 , ◦ 4 ω9 = 150 (Batygin and Brown (2016)). Figures 1 and 2 shows the comparison of the two analytic approaches with the numerical integrations. We 1 notice good agreement, thus, from now on we will consider the angular momentum approach to make 400 600 800 1000 our analysis. a9 (au)

3. Constraining a planet 9 that yield ◦ 5.9 Fig. 3.— Inclination gained by iv4 with respect tilt to the initial reference frame supposed to coincide Figures 3 and 4 show how the inclination of with the current solar equator for different semi- iv4 after 4.5 Gy depends on orbital elements of major axes and initial inclinations of pl9 assuming × −5 the perturbing planet. Here we fixed the mass of a mass of 3 10 M⊙ and an eccentricity of 0.7. −5 The horizontal line stand for the current inclina- pl9 at 3 × 10 M⊙. The largest value of ∆I in each of these figures stands for the case where the tion of iv4 with respect to the solar equator. angular momenta of pl9 and iv4 turn 180◦ around one another. For smaller semimajor axes of pl9, more than one cycle is accomplished by the pair of angular momentum vectors. o The plots in Figs. 3 and 4 allow us to com- I9=30 ∼ ◦ pute pl9 parameters that yield 5.9 inclina- e9=0.2 tion between the solar equator and iv4 after 4.5 e9=0.4 e9=0.6 Gy. for instance, Fig. 3 reveals that a planet 8 e9=0.8 −5 of 3 × 10 M⊙, eccentricity of 0.7 and semimajor axis of 600 au has to have an initial inclination of 6 30◦ relative to the solar equator to cause the ob-

served tilt. Figures 5 and 6 show the loci of pl9 I (deg.) ∆ 4 semimajor axis and eccentricity that yield a tilt of 5.9◦ of iv4 relative to the solar equator for four possible masses for the distant planet and two ini- 2 tial values of pl9 inclination. We notice that for −5 ◦ a mass 2.5 × 10 M⊙ and I9 = 30 , there are just a few choices of planets that can yield a 5.9◦ 400 600 800 1000 −5 a9 (au) tilt. A mass as small as 2 × 10 M⊙ is unable ◦ to yield the right tilt for iv4 if I9 = 30 . For Fig. 4.— The same as Fig. 3, but for diﬀerent 1 We also ran one numerical integration with all eight planets eccentricities and semimajor axes of pl9, assuming and conﬁrmed that the orbital plane of the inner ones just ◦ precessed around a common plane, which in this case would an inclination of 30 . be a iv8 very close to iv4.

6 higher inclinations of pl9 it is possible for iv4 to achieve a tilt of 5.9◦ for somewhat smaller masses of the perturber. The constraints on pl9 orbit that we obtain here have similarities with those 0.9 0.9 obtained by Batygin and Brown (2016) using con- siderations on the orbital alignment of TNOs, al- 0.6 0.6 o though we usually determine a higher eccentricity I9=30 for a given semimajor axis. For example, the stan- eccentricity 0.3 0.3 -5 -5 m9 = 2.5*10 MSun m9 = 3*10 MSun dard pl9 in Batygin and Brown (2016) with m9 = 10M and I = 30◦ has a = 700 au and e =0.6. 0 0 ⊕ 9 9 9 300 600 900 300 600 900 In our case, for the same m9, I9 and a9 the eccentricity must be 0.8. In Brown and Batygin 0.9 0.9 ◦ (2016) the best pl9 for I9 = 30 and m9 = 10M⊕

0.6 0.6 has a9 = 600 au and e9 = 0.5. In our analysis for the same I9, m9 and a9 the eccentricity

eccentricity 0.3 0.3 -5 -5 must be 0.71. Comparing with Malhotra et al. m9 = 4*10 MSun m9 = 5*10 MSun (2016) our eccentricities for pl9 are much higher, 0 0 300 600 900 300 600 900 since Malhotra et al. (2016) determined eccentric- semimajor axis (au) semimajor axis (au) ities lower than 0.4 for a9 = 665 au. As iv4 and pl9 orbital planes evolve, their in- Fig. 5.— Semimajor axis vs. eccentricity of pl9s tersections of the solar equator plane deﬁnes a dif- that yield a tilt of 5.9◦ to iv4 with respect to the ◦ ference of longitudes of ascending nodes on that solar equator, for I9 = 30 plane. Figures 7 and 8 show, respectively, for ◦ ◦ I9 = 30 and I9 = 45 , pl9 semimajor axis and longitude of the ascending node (Ω) that yield 5.9◦ for iv4, on the solar equator plane, with respect to iv4 longitude of the ascending node at 4.5 Gy. This is also shown for four possible masses of the distant planet. We see that iv4 and pl9 are on average opposed by 180◦ on the solar equator. This allows 0.9 0.9 us to compute possible directions for the longitude of the ascending node of pl9 on the ecliptic plane. 0.6 0.6 Once we know the inclinations and longitudes I =45o 9 of the ascending node of pl9 and also the inclina-

eccentricity 0.3 0.3 m = 2.5*10-5 M m = 3*10-5 M tion and ascending node of the solar equator with 9 Sun 9 Sun respect to the ecliptic, we can with a couple of 0 0 300 600 900 300 600 900 rotations compute the longitude of the ascending node of pl9 on the ecliptic. Figures 7 and 8 already 0.9 0.9 suggest that pl9’s longitude of node will be constrained in a 180◦ range. On the ecliptic, Figure 9 0.6 0.6 shows the frequency of possible Ω’s of pl9 for four × −5 eccentricity 0.3 0.3 possible I9. All masses of pl9 from 2 10 M⊙ to -5 -5 −5 m9 = 4*10 MSun m9 = 5*10 MSun 5 × 10 M⊙ are included in these plots. The ver- 0 0 tical lines depicts the range of Ω’s determined by 300 600 900 300 600 900 ◦ ± ◦ semimajor axis (au) semimajor axis (au) Batygin and Brown (2016) (113 13 ). Still for ◦ I9 = 30 Fig. 10 shows the semimajor axis and ◦ Fig. 6.— The same as Fig. 5 but for I9 = 45 eccentricity of pl9 whose Ω fall inside the range predicted by Batygin and Brown (2016). We see that our approach does not constrain very well the longitude of the ascending node of pl9, but our de-

7 termination usually includes Batygin and Brown (2016) prediction based on the longitude of the ascending nodes of the distant TNO’s. Interest- ◦ o ingly we have a better match for I9 > 45 . For I9=30 ◦ 270 270 I9 = 25 and lower we do not obtain any over- lapping with the range of node longitudes from

(deg.) 180 180 ◦

Ω Batygin and Brown (2016) work. For I9 = 30

∆ 90 90 we have compatibility with Batygin and Brown -5 -5 × −5 ∼ m9 = 2.5*10 MSun m9 = 3*10 MSun (2016) only for m9 & 4 10 M⊙( 13.3M⊕). This seems to indicate that pl9’s inclination can- 300 600 900 300 600 900 not be much smaller than 30◦ and, if so, it needs a −5 mass of the order of 5 × 10 M⊙(∼ 17M⊕), some- 270 270 what larger than the 10M⊕ usually assumed. This result does not match Malhotra et al. (2016), who

(deg.) 180 180 Ω give two choices for the pair I and Ω , which are

∆ 9 9 90 90 (18◦, 101◦) and (48◦, 355◦). The higher inclination m = 4*10-5 M m = 5*10-5 M 9 Sun 9 Sun is associated with a longitude of the node that can- 300 600 900 300 600 900 not explain the tilt of the giant planets relative to semimajor axis (au) semimajor axis (au) the solar equator (see Fig. 9). Fig. 7.— Semimajor axis vs. Ω of pl9s that yield It must be noted that when we refer to pl9’s 5.9◦ inclination tilt to iv4 with respect to the solar inclination we mean its initial inclination with re- ◦ equator, for I9 = 30 . spect to the solar equator plane which coincided with iv4. The ﬁnal pl9 inclination with respect to the ecliptic which should be compared with current pl9’s inclination will vary a little from the initial reference inclination. For instance, for the ◦ case where I9 = 30 , the ﬁnal pl9’s inclination with respect to the ecliptic will vary in a range from 27.3◦ to 37.2◦. If we restrict to the Ω’s con- o I9=45 strained by Batygin and Brown (2016) this range 270 270 shrinks to 29.7◦ to 33.2◦.

(deg.) 180 180 Ω

∆ 4. Conclusions 90 90 -5 -5 m9 = 2.5*10 MSun m9 = 3*10 MSun Some ideas had been put forward to possibly 300 600 900 300 600 900 explain the inclination of the invariant plane of the known planets relative to the solar equatorial plane (Batygin et al. 2011) but in view of the 270 270 convincing case presented in Batygin and Brown

(deg.) 180 180 (2016) for the existence of pl9, it is quite nat- Ω

∆ ural to suppose that such a tilt was caused by 90 90 -5 -5 m9 = 4*10 MSun m9 = 5*10 MSun a slow precession of iv4 around the total angular momentum vector of the solar system (in- 300 600 900 300 600 900 semimajor axis (au) semimajor axis (au) cluding pl9). In this paper we constrained possible masses and orbital elements for pl9 that Fig. 8.— Semimajor axis vs. Ω of pl9s that yield can account for the present tilt of iv4 with the 5.9◦ inclination tilt to iv4 with respect to the solar solar equator. Our results are usually compati- ◦ equator, for I9 = 45 ble with those of Batygin and Brown (2016) and Brown and Batygin (2016) though with somewhat larger eccentricities. We also determine a range

8 of possible longitudes of the ascending node for 1 1 o o pl9 which often overlaps with the range given I =25 I =30 9 9 in Batygin and Brown (2016) except for smaller masses and inclinations of pl9. For instance, for ◦ 0.5 0.5 I9 = 30 we need a mass larger than ∼ 4 ×

frequency −5 10 M⊙ ∼ 13M⊕ to match the range of the longitudes of the ascending node for pl9 proposed by 0 0 Batygin and Brown (2016). 90 180 270 90 180 270

1 1 o o Acknowledgments I9=45 I9=60 R.D. acknowledges support provided by grants 0.5 0.5 #2015/18682-6 and #2014/02013-5, S˜ao Paulo frequency Research Foundation (FAPESP) and CAPES.

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